text: The potential theory underwent many refinements and changes throughout the years since its first report.  One major theories of note that was developed using Polanyi's theory was the Dubinin theories, Dubinin–Radushkivech and Dubinin–Astakhov equations.
  Using the adsorption potential, the degree of filling of the adsorption space, , can be calculated as
  where  is value of adsorption at temperature T and equilibrium pressure p,  is the maximum value of adsorption, and  is the characteristic energy of adsorption in kJ/mol,  is the loss in Gibbs free energy in adsorption equal to  and  is the fitting coefficient.  The Dubinin–Radushkivech equation where  is equal to 2 and the optimized Dubinin-Astakhov equation where  is fit to experimental data can be simplified to
  Other studies have used the Dubinin–Astakhov in a similar form of 
  where  is equilibrium adsorbed concentration of adsorbent in mg/g,  is maximum adsorbed concentration of adsorbent in mg/g,  is the effective adsorption potential, where equal to ,  is equilibrium concentration of adsorbent in the solution phase in mg/L, and  is the adsorbent solubility in water in mg/L.
  The characteristic energy of adsorption can be related to a characteristic energy of adsorption for a standard vapor on the same surface, , through the use of an affinity coefficient, 
  The affinity coefficient is a ratio of the properties of the sample and standard vapors
  where  and  are the polarizabilities of the sample and standard vapors, respectively.  Many studies have been performed to determine optimal fitting coefficients, , and affinity coefficients, , to best describe the adsorption of gases and vapors onto solids.  As a result, the Dubinin–Astakhov equation remains in use in adsorption studies due to the accuracy it can obtain when fitted with experimental results.